Anisotropic Compact Objects in Modified f(R,T) gravity
aa r X i v : . [ phy s i c s . g e n - ph ] S e p Anisotropic Compact Ob jects in Modified f ( R, T ) gravity S Dey , A Chanda , B C Paul Department of Physics, University of North Bengal, Siliguri, Dist. Darjeeling 734013, West Bengal, IndiaIUCAA Centre for Astronomy Research and Development, North BengalE-mail : [email protected], [email protected], [email protected] Abstract.
We obtain a class of anisotropic spherically symmetric relativisticsolutions of compact objects in hydrostatic equilibrium in the f ( R, T ) = R + 2 χT modified gravity, where R is the Ricci scalar, T is the trace of the energy momentumtensor and χ is a dimensionless coupling parameter. The matter Lagrangian is L m = − (2 p t + p r ), where p r and p t represents the radial and tangential pressures.Compact objects with dense nuclear matter is expected to be anisotropic. Stellarmodels are constructed for anisotropic neutron stars working in the modified Finch-Skea (FS) ansatz without preassuming an equation of state. The stellar models areinvestigate plotting physical quantities like energy density, anisotropy parameter, radialand tangential pressures in all particular cases. The stability of stellar models arechecked using the causality conditions and adiabatic index. Using the observed massof a compact star we obtain stellar models that predicts the radius of the star and EoSfor matter inside the compact objects with different values of gravitational couplingconstant χ . It is also found that a more massive star can be accommodated with χ < f ( R, T ) gravity. ompact Objects in f ( R, T ) gravity
1. Introduction
General theory of Relativity (GTR) is a geometric theory of gravitation formulated onthe concept that gravity manifests itself as the curvature of space-time. Although GTRis a fairly successful theory at low energy, it is entangled with some serious issues at ul-traviolet and infrared limits. Some of the astronomical observational evidences namely,Galactic, extra Galactic and cosmic dynamics are not understood in the framework ofGTR. The needle of hope points to the concept of the existence of exotic matter thatrepresents the dark energy [1, 2] which we need if matter sector of GTR is to be mod-ified. On the other hand a modification of the gravitational sector to fit the missingmatter-energy of the observed universe is also another important area of present re-search. In the literature [3, 4, 5, 6] a number of theories of gravity with modification ofthe gravitational sector came up to understand the evolution of the observed universeas well as to solve some of the issues of non- renormalizability [7, 8] in GTR. In 1970,Buchdahl [9] using a non-linear function of Ricci scalar namely, f ( R ) gravity theoryfirst introduced a modification of theory of gravity to explain some of the drawbacks inFriedmann- Lemaˆıtre-Robertson-Walker cosmological models. A higher derivative termin the gravitational action in the form R -term was considered by Starobinsky [10] andfound the existence of inflationary solutions in cosmology.Recently, Harko and his collaborators [11] introduced a more generalized form of grav-ity, the f ( R ) -gravity which consists of a self-assertive expression of the Ricci scalar ( R )and the trace of the energy-momentum tensor ( T ) together introducing f ( R, T )-theoryof gravity. The modified theory is interesting as it is effectively accommodate the latetime acceleration of the universe. Consequently, there is a spurt in research activitiesin understanding astrophysical objects of interest in the modified theory of gravity. Itis known that the presence of an extra force perpendicular to the four velocity in the f ( R, T ) gravity helps test particles to follow a non-geodesic trajectory. It is shown [12]that for a specific linear form of f ( R, T ) -theory, say f ( R, T ) = R + f ( T ), the trajec-tory of the particles become a geodesic path. It is known [13] that that the f ( R, T )theory of gravity pass solar system test satisfactorily. A number of cosmological models[15, 16, 17] are constructed in the f ( R, T ) theory of gravity which accommodates theobserved universe successfully. Consequently, Moraes et al. [18] studied the equilibriumconfiguration of quark stars with MIT bag mode. It is shown [19] that an analyticalstellar model for compact star in f ( R, T ) gravity may be obtained considering a correctform of the Tolman-Oppenheimer-Volkoff (TOV) equation. Deb et al. [20] analyzedboth isotropic and anisotropic spherically symmetric compact stars and presented thegraphical analysis of LMC X-4 star model. The effect of higher curvature terms presentin f ( R, T ) gravity is probed in compact objects [21] making use of EoS given by poly-tropic and MIT bag model. The physical properties of a star in the above case canbe derived knowing EoS, i.e. , p = p ( ρ ) which is not yet known for a compact objectat extreme terrestrial condition. In the absence of a reliable information of the EoSat very high densities, assumption of the metric potentials, based on the geometry has ompact Objects in f ( R, T ) gravity f ( R, T ) gravity and construct stellar models. FS metricoriginated to correct the Dourah and Ray [26] metric which is not suitable for compactobject, Finch and Skea [27] modified the metric to describe relativistic stellar models.Subsequently, FS metric with a modification in 4- dimensions [28, 29, 30] and in higherdimensions [31, 32, 33] are considered to explore astrophysical objects. In compactobjects the interior pressure may not be same in all directions, thus the study of thebehaviour of anisotropic pressure for a spherically symmetric stellar model is importantto explore. Ruderman [35] shown that at high density ( > g/cm ) nuclear matterobject may be treated relativistically which exhibits the property of anisotropy. Thereason for incorporating anisotropy is due to the fact that in the high density regimeof compact stars the radial pressure ( p r ) and the transverse pressure ( p t ) are not equalwhich was pointed out by Canuto [36]. There are other reasons to assume anisotropyin compact stars which might occur in astrophysical objects for various reasons namely,viscosity, phase transition, pion condensation, the presence of strong electromagneticfield, the existence of a solid core or type 3A super fluid, the slow rotation of fluids etc.In this paper we construct relativistic stellar models and predict EoS in the frameworkof a linear f ( R, T ) gravity with isotropic or anisotropic fluid distribution.The outline of the paper is as follows: in section we present the basic mathematicalformulation of f ( R, T ) theory and the field equations. In section , a class of relativisticsolutions are obtained for different parameters of the theory. In section the constraintsto obtain stellar models are presented. In section , general properties of compact stars,the stability of stellar models, energy conditions, mass to radius etc. are discussed. TheEoS of mater inside the star is also predicted. Finally, we discuss the results in section .
2. The Gravitational action and the field equations in f ( R, T ) gravity The gravitational action for modified theory of gravity is given by S = 116 π Z f ( R, T ) √− g d x + Z L m √− g d x, (1)where f ( R, T ) is an arbitrary function of the Ricci scalar ( R ) and ( T ) is the trace ofthe energy-momentum tensor T µν . The determinant of the metric tensor g µν is given by g and L m is the Lagrangian density of the matter part. We consider gravitation unit c = G = 1. The field equations for the modified gravity theory can be obtained byvarying the action S with respect to the metric tensor g µν which is given by,( R µν − ∇ µ ∇ ν ) f R ( R, T ) + g µν ✷ f R ( R, T ) ompact Objects in f ( R, T ) gravity − g µν f ( R, T ) = 8 π T µν − f T ( R, T ) ( T µν + Θ µν ) , (2)where f R ( R, T ) denotes the partial derivative of f ( R, T ) with respect to R , and f T ( R, T )denotes the partial derivative of f ( R, T ) with respect to T . R µν is the Ricci tensor, ✷ ≡ √− g ∂ µ ( √− g g µν ∂ ν ) is the D’Alembert operator and ∇ µ represents the co-variantderivative, which is associated with the Levi-Civita connection of the metric tensor g µν .The energy momentum tensor T µν for perfect fluid changes the role in the f ( R, T )-modified gravity because of the presence of ∇ µ ∇ ν R and ( ∇ µ R )( ∇ ν R ) and terms whichoriginate from trace of the energy momentum tensor T in the field equation. In thepaper, we consider compact objects with anisotropic matter distribution in the modifiedgravity. The stress-energy tensors T µν and Θ µν are defined as, T µν = g µν L m − ∂L m ∂g µν , (3)Θ µν = g αβ δT αβ δg µν . (4)Using eq. (2) the covariant divergence of the stress-energy tensor can be written as ∇ µ T µν = f T π − f T (cid:20) ( T µν + Θ µν ) ∇ µ lnf T + ∇ µ Θ µν − g µν ∇ µ T (cid:21) . (5)It may be mentioned here that the covariant derivative of the stress-energy tensor in f ( R, T ) theory does not vanishes, which is different from the f ( R )-theory. Consequentlywe describe an effective energy density and pressure which however leads T effµν ; µ = 0.In the modified gravity f ( R, T ) = R + 2 χT , where χ is a coupling constant, thefield eq. (2) can be represented as G µν = 8 π T effµν (6)where G µν is the Einstein tensor and T effµν is the effective energy-momentum tensor.The energy-momentum tensor for anisotropic matter distribution is given by T µν = ( ρ + p t ) u µ u ν − p t g µν + ( p r − p t ) v µ v ν (7)where v µ is the radial four-vector, while u ν is four velocity vector, ρ , p r and p t arethe energy density, the radial and tangential pressures respectively. Here, we considerthe matter Lagrangian L m = − P , where P = (2 p t + p r ). For anisotropic fluidΘ µν = − T µν − g µν P , the effective energy-momentum tensor becomes T effµν = T µν (cid:16) χ π (cid:17) + g µν χ π ( T + 2 P ) . (8)The above expression contains the original matter stress-energy tensor T µν and thecurvature terms [11]. We consider f ( R, T ) = R + 2 χT and the eq.(5) becomes ∇ µ T µν = − χ π + χ ) ( g µν ∇ µ T + 2 ∇ µ ( g µν P )) . (9)Now the effective conservation of energy equation is given by ∇ µ T effµν = 0 . (10) ompact Objects in f ( R, T ) gravity f ( R, T )- theory of gravity which is an extension of both GTR and f ( R )-gravity.
3. Modified Field Equations in f ( R, T ) gravity We consider a spherically symmetric metric for the interior spacetime of a static stellarconfiguration given by ds = e ν ( r ) dt − e λ ( r ) dr − r ( dθ + sin θdφ ) , (11)where ν and λ are the metric potentials which are functions of radial coordinate ( r )only. The non-zero components of the energy momentum tensors are given by T = ρ ( r ) , (12) T = − p r ( r ) , (13) T = T = − p t ( r ) (14)where p r and p t are radial and tangential pressures respectively. Using eqs. (6) - (8),the field equations can be rewritten as e − λ (cid:16) ν ′ r + 1 r (cid:17) − r = 8 πp effr , (15) e − λ (cid:16) ν ′′ + ν ′ + ν ′ − λ ′ r − ν ′ λ ′ (cid:17) = 8 πp efft , (16) e − λ (cid:16) λ ′ r − r (cid:17) + 1 r = 8 πρ eff , (17)where the prime ( ′ ) is differentiation w.r.t. radial coordinate, ρ eff , p effr and p efft are theeffective density, radial pressure and tangential pressure. We get ρ eff = ρ + χ π (9 ρ − p r − p t ) (18) p effr = p r − χ π (3 ρ − p r − p t ) (19) p efft = p t − χ π (3 ρ − p r − p t ) . (20)To study the matter content inside the compact objects, the field eqs.(15)-(17) are usedto determine the components of T µν i.e. ρ , p r and p t . Using eqs.(15) and (16) we get asecond order differential equation which is ν ′′ + ν ′ − ν ′ λ ′ − λ ′ r − ν ′ r − r + e λ r = 2(4 π + χ ) ∆ e λ (21)where ∆ = p t − p r , which represents the measure of anisotropy in pressure. In terms ofeffective pressures we get p effr − p efft = (cid:16) χ π (cid:17) ∆ ompact Objects in f ( R, T ) gravity i.e. p r = p t one finds isotropy in the effective pressure. It is also noted that the effectivepressure difference vanishes even if p r = p t when χ = − π .In this section we adopt following transformations first proposed by Durgapal andBannerji [37] on the matric potentials to obtain relativistic solutions A y ( x ) = e ν ( r ) , Z ( x ) = e λ ( r ) , x = Cr . where A and C are arbitrary constants. The above transformation reduces eq. (21) toa second order differential equation which is given by4 x z ¨ y + 2 x ˙ z ˙ y + y (cid:20) x ˙ z − z + 1 − π + χ ) C x ∆ (cid:21) = 0 (22)where the overdot denotes differentiation w.r.t. the variable x . The eq. (22) is further simplified introducing Z ( x ) [38] as Z = 11 + x . (23)Note that the choice of Z is a sufficient condition for a static perfect fluid sphere whichis regular at the center [39]. Eq.(22) can be expressed as4(1 + x ) ¨ y − y + (1 − α ) y = 0 (24)where α = x +1) (4 π + χ ) Cx . The measure of anisotropy is given by∆ = α x C π + χ )( x + 1) (25)for χ = − π and C = 0. For α = 0, one recovers Finch-Skea model with an isotropicpressure distribution. For anisotropic star, ∆ vanishes at the center ( i.e. p r = p t ), butaway from the centre it is a regular solution which grows showing different patternsof evolution for both the pressures. For − < α <
1, we substitute the following : X = 1 + x and y ( x ) = Z for simplicity in eq.(24) which yields4 X d ZdX − dZdX + (1 − α ) Z = 0 . (26)Once again we introduce the following transformations: Z = w ( X ) X n and u = X γ ,where γ and n are real numbers. The above differential equation can be reduced to astandard Bessel equation. For γ = and n = , the eq. (26) reduces to u d wdu + u dwdu + (cid:20) (1 − α ) u − (cid:21) w = 0 . (27)Now we consider further transformation from u to v variable as (1 − α ) u = v in theeq.(27) which leads to a second order differential equation as follows v d wdv + v dwdv + " v − (cid:18) (cid:19) w = 0 . (28) ompact Objects in f ( R, T ) gravity . The general solution is given by w = c J ( v ) + c J − ( v )where c and c are integration constants, J ( v ) and J − ( v ) are the Bessel functions,which can be written in terms of trigonometric functions. The general solution of theeq.(24) for modified FS-metric in four dimension [39] is given by y ( x ) = (1 − α ) − [( b − a p (1 + C r )(1 − α ))cos p (1 + C r )(1 − α ) + ( a + b p (1 + C r )(1 − α ))sin p (1 + C r )(1 − α )] (29)where, a = c q π and b = − c q π are arbitrary constants of the metric. We considerthe metric potential of the modified 4-dimensional FS -metric as e λ ( r ) = 1 + C r , (30) e ν ( r ) = (1 − α ) − A [( b − a p (1 + C r )(1 − α ))cos p (1 + C r )(1 − α ) + ( a + b p (1 + C r )(1 − α ))sin p (1 + C r )(1 − α )] (31)where C , a , b , A and α are the five unknowns. For α = 0, the Finch- Skea solutionobtained in GR for 4 - dimensions with isotropic fluid is recovered [27]. The relativisticsolution for − < α < a , b and C forgiven values of α and χ . It may be mentioned here that for α ≥
1, the stellar modelsare not stable. Consequently, we consider − < α < f ( R, T )- modified gravityto construct stellar models for compact objects.
4. Analysis for Stellar Models
The following conditions [43] are imposed on the relativistic solutions for a physicallyrealistic stellar configurations for compact objects in the modified gravity : • At the boundary of a static star ( i.e. at r = b ), the interior space-time is matchedwith the exterior Schwarzschild solution. For the continuity of the metric functions atthe surface, one consider e ν ( r ) | r = b = (cid:16) − Mb (cid:17) (32) e λ ( r ) | r = b = (cid:16) − Mb (cid:17) − (33) • The radial pressure ( p r ) drops from its maximum value (at the center) to vanishingvalue at the boundary , i.e. , at r = b , p ( r = b ) = 0, the radius of the star b can be ompact Objects in f ( R, T ) gravity • The causality condition is satisfied when the speed of sound v = dpdρ • The gradient of the pressure and energy-density should be negative inside the stellarconfiguration, i . e . , dp r dr < dρdr < • At the center of the star, ∆(0) = 0 which implies zero radial and tangential pressure, p r (0) = p t (0). • The anisotropic fluid sphere must satisfy the following three energy conditions, viz., (a)null energy condition (NEC), (b) weak energy condition (WEC) and (c) strong energycondition (SEC) if it is made up of normal fluid. • The adiabatic index : Γ = ρ + pp dpdρ > required for ensuring stability of the stellarconfiguration [40].There are three field equations and five unknowns, to solve the equations two adhoc assumptions are necessary for obtaining exact solutions. Thus to construct stellarmodels, the unknown metric parameters a , b , C for a given mass ( M ) and radius ( r = b )of a star are to be determined from the boundary conditions making use of permissiblevalues of α and χ for a realistic stellar model. Alternatively, for a given mass we canpredict the radius of the compact objects for values of the other parameters.
5. Physical Properties of compact stars for − < α < − < α <
1. Asthe relativistic solutions are highly complex we analyze numerically the variations ofthe energy density, radial pressure, transverse pressures, energy conditions, anisotropyof pressure and stability for a given value of the model parameters. The graphical plotsare important for predicting the EoS of the observed compact objects. We consideruncharged anisotropic stellar objects.
Density and Pressure of a compact objects in f ( R, T ) gravity In the f ( R, T ) - modified gravity we determine physical parameters, namely, energydensity ( ρ ), radial pressure ( p r ) and tangential pressure ( p t ). The metric potentials e λ ( r ) and e ν ( r ) given by eqs. (30) and (31) are employed in eqs. (15) - (17) to determine theenergy density ( ρ ), radial pressure ( p r ) and tangential pressure ( p t ) which are given by ρ = C ( sin ( √ h )( a h + bh o √ h ) +cos ( √ h )( b h − a h o √ h )) g ( r,a,b,C,χ ) , (34) p r = C (cid:0) h cos √ h − h sin √ h (cid:1) g ( r, a, b, C, χ ) , (35) p t = C (cid:0) h sin √ h + h cos √ h (cid:1) g ( r, a, b, C, χ ) . (36)where the denominator is denoted as g ( r, a, b, C, χ ) = 12 ( χ + 6 πχ + 8 π ) ( Cr + 1) (sin √ h (cid:0) a + b √ h (cid:1) +cos √ h (cid:0) b − a √ h (cid:1) ), ompact Objects in f ( R, T ) gravity H km L Ρ H k m - L Figure 1.
Radial variation of energy-density ( ρ ) in PSR J0348+0432 for χ = 1 (Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) (considering α = 0 . H km L p r H k m - L Figure 2.
Radial variation of radial pressure ( p r ) in PSR J0348+0432 for χ = 1(Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) for α = 0 . h o = χ ( Cr ( α + 3) + 12) + 12 π ( Cr + 3), h = − ( α − Cr + 1), h = χ ( − Cr ( α − − α − π ( Cr + 3), h = a √ h ( Cr ( α + 3) χ + 12 π ( Cr + 1)) + b ( χ (2 Cr (3 − α ) − α + 9) − π (2 α − Cr + 1)), h = a ( χ (2 Cr (5 α −
3) + 9( α − π (2 α − Cr + 1)) + b √ h ( Cr ( α + 3) χ + 12 π ( Cr + 1)), h = a ( χ (2 Cr (3 − α ) − α + 9) − π ( Cr ( α −
1) + 2 α − b √ h ( Cr (5 α − χ + 12 π ( Cr ( α − − h = a √ h ( Cr (3 − α ) χ − π ( Cr ( α − − b ( χ (2 Cr (3 − α ) − α + 9) − π ( Cr ( α −
1) + 2 α − ρ ), radial pressure ( p r ) and tangentialpressure ( p t ) are plotted for PSR J0348+0432 in Figs. (1), (2) and (4) respectivelyfor α = 0 . χ . It is evident that the physical quantities are maximumat the origin which however, decrease monotonically away from the centre. As χ isincreased the values of the physical parameters decreases. Similarly, the radial variationof radial pressure ( p r ) and tangential pressure ( p t ) for different α for χ = 1 are plottedin Figs. (3) and (5) respectively. It is noted that as α increases, the radial pressures andtangential pressure decreases which are positive and regular at the origin with maximumvalues. Thus the model is free from physical and mathematical singularities. It is also ompact Objects in f ( R, T ) gravity H km L p r H k m - L Figure 3.
Radial variation of radial pressure ( p r ) in PSR J0348+0432 for α = 0(Gray), α = 0 . α = 0 . α = 0 . α = 0 . α = 0 . χ = 1 H km L p t H k m - L Figure 4.
Radial variation of transverse pressure ( p t ) in PSR J0348+0432 for χ = 1(Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) for α = 0 . evident that the radial variation of energy density gradient and radial pressure gradientfor different values of χ are plotted in Figs. (6) and (7) respectively, which are foundnegative and it increases as χ is decreased for α = 0 . Anisotropic Star
Th anisotropy of a compact star which is determined by the difference of tangential andradial pressures is obtained from eqs. (35) and (36) as follows:∆ = p t − p r = C r α χ + 4 π ) ( Cr + 1) . (37)An isotropic stellar model can be obtained for α = 0 in 4-dimensions, it is evident thatin modified gravity it always permits anisotropic star unless χ = − π which followsfrom eq. (37). In GTR, it is known that FS metric does not permit anisotropic compactstar in a 4-dimensional geometry, but recently it is shown that a higher dimensionalextension of the Finch-Skea geometry permits an anisotropic star [32]. As the structureof f ( R, T ) -gravity is interesting found that anisotropic star is always permitted in a ompact Objects in f ( R, T ) gravity H km L p t H k m - L Figure 5.
Radial variation of transverse pressure ( p t ) in PSR J0348+0432 for α = 0(Gray), α = 0 . α = 0 . α = 0 . α = 0 . α = 0 . χ = 1 - - - - H km L d Ρ d r H k m - L Figure 6.
Radial variation of energy-density gradient ( dρdr ) in PSR J0348+0432 for χ = 1 (Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) for α = 0 . χ values with a given α . It is found that for χ >
0, ∆ is positive i.e. , p t > p r which in turnimplies that the anisotropic stress is directed outwards, hence there exists a repulsivegravitational force that allows the formation of super massive stars.The radial variation of ∆ for different values of α in the range ( − . to + 0 . χ = 1. It is evident that when χ = 0 it corresponds toisotropic star ( α = 0) in four dimensions but when α < < χ with | χ | 6 = 4 π .The anisotropy increases as α increases but it decreases if α ismore negative. It is also noted that for negative values of α one gets ∆ <
0, where theradial pressure is greater than the tangential one, i.e. , p r > p t . We also note that when α = − .
5, we get a situation where ∆ < − > χ > − . < α = − .
2, in the range − > χ > − . α = − .
0, in the range − > χ > − .
4. Thus it is clear that this negative range of χ varies with negative α values which permits compact objects with p t > p r . However,our model is not allowed for negative χ values with positive α . ompact Objects in f ( R, T ) gravity - - - ´ - H km L dp d r H k m - L Figure 7.
Radial variation of pressure gradient ( dp r dr ) in PSR J0348+0432 for χ = 1(Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) for α = 0 . H km L D H k m - L Figure 8.
Radial variation of anisotropy parameter (∆) in PSR J0348+0432 for χ = 1(Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) for α = 0 . - - - - H km L D H k m - L Figure 9.
Radial variation of anisotropy parameter (∆) in PSR J0348+0432 for α = − . α = − . α = − . α = − . α = − . α = 0(Gray), α = 0 . α = 0 . α = 0 . α = 0 . α = 0 . χ = 1 ompact Objects in f ( R, T ) gravity H km L v r Figure 10.
Radial variation of v r in PSR J0348+0432 for χ = 1 (Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) (considering α = 0 . H km L v t Figure 11.
Radial variation of v t in PSR J0348+0432 for χ = 1 (Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) (considering α = 0 . Stability of the Stellar Model
Herrera cracking concept
The stability of a stellar model is studiednumerically plotting the radial variation of the square of the radial speed of sound( v r = dp r dρ ) and square of the transverse speed of sound ( v t = dp t dρ ) separately in Figs.(10) and (11) respectively. It is found that a stable configuration of anisotropic compactobject can be accommodated. Herrera and Abreu [41] pointed out that for a physicallystable stellar system made of anisotropic fluid distribution the difference of square ofthe sound speeds should maintain its sign inside the stellar system. Accordingly, in apotentially stable region, square of the radial sound speed should be greater than thesquare of the tangential sound speeds. Hence, according to Herreras cracking conjecturethe required condition | v t − v r | ≤ | v t − v r | w.r.t. r in Fig. (12) and it is found that the condition is found to satisfy | v t − v r | ≤ χ with α = 0 . Adiabatic index
The stiffness of the EoS for given energy density ischaracterised by adiabatic index which has significant importance for understanding the ompact Objects in f ( R, T ) gravity H km L È v t - v r È Figure 12.
Radial variation of | v t − v r | in PSR J0348+0432 for χ = 1 (Red), χ = 3(Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) (considering α = 0 . stability of relativistic as well as non-relativistic compact objects. Chandrasekhar beganthe study of the dynamical stability against infinitesimal radial adiabatic perturbationof the stellar system. It is estimated that the magnitude of the adiabatic index shouldbe greater than in the interior of a dynamically stable stellar object. For anisotropicfluid distribution the adiabatic index is given by,Γ = ρ + p r p r dp r dρ . (38)The radial variation of the adiabatic index is plotted in Fig. (13) for different values of χ . The stellar models obtained here are found to have dynamical stability as Γ ≥ .The stellar models are stable against infinitesimal radial adiabatic perturbations. InFig. (14) we plot the radial variation of adiabatic index (Γ) for different values of theparameter 0 < α < . χ = 1. We note acceptable range 0 < α < . χ = 0 . < α < . χ = 1 , . .
2. Thus on increasing χ the acceptable range of α remains same for anisotropic star. Thus in f ( R, T ) modified gravity we get an upperbound on α for χ > Energy conditions of the stellar model in the f ( R, T ) gravity The energy conditions play a crucial role in determining the observe normal or exoticnature of matter inside the stellar model. The energy conditions are null (NEC),dominant (DEC), strong (SEC) and weak energy conditions (WEC). in an anisotropicfluid distribution are expressed as follows:
NEC : ρ ≥ , (39) WEC1 : ρ + p r ≥ , WEC2 : ρ + p t ≥ , (40) SEC : ρ + p r + 2 p t ≥ , (41) DEC1 : ρ − p r ≥ , DEC2 : ρ − p t ≥ , (42) ompact Objects in f ( R, T ) gravity H km L G Figure 13.
Radial variation of Γ in PSR J0348+0432 for χ = 1 (Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) (considering α = 0 . H km L G r Figure 14.
Radial variation of Γ in PSR J0348+0432 for α = 0 . α = 0 . α = 0 . α = 0 . α = 0 . χ = 1) The evolution of all the energy conditions against the radial coordinate r for the compactstellar structure is studied here for different χ with α = 0 . f ( R, T )-gravity. Theseare shown graphically in the Figs. (15)- (19).
Stellar Mass - Radius Relation
For a static spherically symmetric stellar models with anisotropic fluid Buchdahl founda limit on the mass to radius ratio, i . e . MR < [42]. In this section we analyzegraphical behaviour of the mass- radius relation for different values of the parameters.The effective mass is given by m ( r ) = Z r πr ′ ρdr ′ . (43)We consider PSR J0348+0432 with observed mass equal to M = 2 . ± M ⊙ . Nowplotting the observed mass in the mass-radius curve in Fig.(20), it is found that one canpredict the variation of the size of a compact object for different χ values. In Fig.(20) itis shown that for a given mass of known object, the radius increases for the increasingvalues of χ , thus the compactness factor of the star decreases. Thus, we can state that ompact Objects in f ( R, T ) gravity χ we can find more dense object comparatively. The mass functionis regular at the center of the compact stellar structure. As it is not yet measured theradius of a star accurately, many aspects of a compact object may be understood oncethe mass and radius are determined accurately. Class of Stellar Models with EoS
The different physical parameters a , b , C of Finch-Skea metric given by eqs. (35)and (36) are determined using the boundary conditions, satisfying the criterion for aphysically realistic stellar object. We tabulated different metric parameters in Tables-1and 2 for PSR J0348 + 0432 which admits different class of stellar model in f ( R, T )gravity. The value of C is calculated for a particular stellar object which is independentof χ and α . Considering C = 0 . M = 2 . ± M ⊙ and radius, R = 11 km . In Table-1,values of a and b for different χ at α = 0 . a and b are displayed for different α taking χ = 1. We also tabulated parameters for differentknown sources namely, Vela X-1, 4U 1820-30, Cen X-3, LMC X-4, SMC X-1 with theirprecise estimated mass. For χ = 1 and α = 0 . χ and α are taken differentthen for a given mass one estimates the radii which is different from the estimated valuein the Table-2. As the radius of a star can not be measured precisely, we can predictthe radius in the models. The predicted radii in the modified gravity with FS-geometrypermits very compact objects namely, neutron stars, strange stars. Equation of State
The variation of the energy-density and radial pressure are plotted in Figs. (1) and (2)from which we determine functional form by best fitting the curve. Here we determinethe best fit relation between ρ and p r , the expressions so obtained for different χ valueshave been listed in Table-1 for a given α . In Fig.(21) we plot p r vrs. ρ for PSRJ0348+0432 with different χ for a given α . It is found that EoS for PSR J0348+0432is non-linear for χ = 1 and χ = 3, but linearity develops in Fig. (21) as the values of χ increases. It is shown that quadratic fitting of the EoS curve is better than a linear onefor lower value of χ . Thus the MIT Bag model representing the EoS in a compact staris not suitable in a compact object with FS geometry, it predicts a non-linear EOS.
6. Discussion
In the paper we obtain a class of relativistic solutions for compact objects in hydro-static equilibrium in a modified gravity f ( R, T ) = R + 2 χT . Since the field equationsare highly complex we adopt a technique to project the field equation in a second orderdifferential equation. The anisotropic stellar models are constructed here. We analyze ompact Objects in f ( R, T ) gravity Figure 15.
WEC 1
Figure 16.
WEC 2
Figure 17.
DEC 1
Figure 18.
DEC 2
Figure 19.
SEC r H km L m H M Q L Figure 20.
Mass - Radius relation in PSR J0348+0432 for χ = − χ = − α = − . χ = 0 (Brown) χ = 1 (Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) for α = 0 . ompact Objects in f ( R, T ) gravity χ b a EoS1 0.21193 0.31475 p r = − . . ρ − . ρ p r = − . . ρ − . ρ p r = − . . ρ p r = − . . ρ
10 0.24515 0.17649 p r = − . . ρ Table 1.
Physical parameters for PSR J0348+0432 in f ( R, T ) gravity for α = 0 . α b a α b a0.1 0.26914 0.28295 - 0.1 0.28035 0.301210.2 0.25967 0.27915 - 0.3 0.28273 0.327540.3 0.24723 0.28104 - 0.5 0.27767 0.357650.4 0.23148 0.29136 - 0.7 0.26633 0.389090.5 0.21193 0.31475 -1.0 0.23950 0.43558 Table 2.
Anisotropy and Metric parameters for PSR J0348+0432 in modified gravityfor χ = 1. Stars Mass ( M ⊙ ) b a C b (km.)Vela X-1 1.77 ± ± ± ± ± Table 3.
Numerical values of physical parameters for different compact object for α = 0 . χ = 1 where b represents the predicted radius of the pulsars. Ρ H km - L p r H k m - L Figure 21.
EoS in PSR J0348+0432 for χ = 1 (Red), χ = 3 (Blue), χ = 5 (Green), χ = 7 (Purple) and χ = 10 (Black) for α = 0 . ompact Objects in f ( R, T ) gravity χ = 0 it corresponds to GR and itrepresents isotropic stellar configuration [27]. In the modified gravity it is found thatstellar models represent anisotropic uncharged compact objects always unless χ = − π .It is also found that realistic stellar models are permitted for a given range of valuesof anisotropy − < α < +0 . i ) The radial variation of energy density, radial pressure and tangential pressureplotted in Figs. (1) - (4) show that they are maximum at the origin which howeverdecrease away from the centre. The radius of the star is determined from the condi-tion that the radial pressure vanishes at the surface. The coupling parameter χ in thegravitational action is playing an important role to accommodate anisotropic compactobjects. We note that both the central density and pressure decreases as χ is increased.We note that as χ is decreased it accommodates a more dense star.( ii ) There is no physical and mathematical singularities as the radial variation ofthe radial pressure ( p r ) and tangential pressure ( p t ) shown in Fig. (2) - (5) are positiveand regular at the origin.( iii ) An isotropic stellar configuration is obtained for α = 0 in eq. (31). For PSRJ0348+0432, the radial variation of ∆ for different χ in Fig.(8) shows that ∆ > i.e. p t > p r for χ > iv ) For χ = 1 the plot of radial variation of ∆ for different values of α in Fig.(9) shows that it admits stellar models with p r > p t indicating the formation of ultracompact objects. It is also noted that the range 0 . < α < f ( R, T ) gravity, the anisotropy is small near the centre whichhowever attains maximum value at the surface. For PSR J0348 + 0432 we determine thevalues of chi for which ∆ <
0. It is found that (i) α = − . − . < χ < − α = − . − . < χ < − α = − .
2, in the range − . < χ < − α is decreased the lower value of χ is increased.( v ) The radial variation of the adiabatic index Γ plotted in Fig.(13) shows that the ompact Objects in f ( R, T ) gravity χ > . A class of relativisticsolutions are obtained here for anisotropy lying in the range 0 < α < . f ( R, T )-gravity with modifiedFS-metric ansatz permits anisotropic star in four dimensions, it is different from thatof GTR result where it accommodates stars with isotropic pressure. We obtain upperbounds on anisotropy α for different χ for anisotropic stars.( vi ) All the energy conditions, viz. , (a) Null energy condition (NEC), (b) Weakenergy condition (WEC) and (c) Strong energy condition (SEC) drawn in Figs. (15) -(19) are satisfied. Thus no exotic matter required for building stellar models.( vii )The mass-radius relation of PSR J0348+0432 plotted in Fig.(20) for differentvalues of χ shows that for a given mass of the compact object, the radius increases foran increasing value of χ . Thus for lower χ , the models accommodates very compactobject as the compactness factor (cid:0) Mb (cid:1) increases (where b is the radius of a star).( viii ) We constructed stellar models for PSR J0348 + 0432 without pre-assumingEoS. Instead we assume modified FS metric ansatz to determine EOS with the metriccoefficients a , b and gravitational coupling parameter χ for an anisotropic configurationwith α = 0 .
5. The probable EoS are tabulated in Table-1, it is evident that both linearand quadratic EoS are obtained. The numerical fitting of the pressure and density curvesshow that the goodness of fit for the quadratic fitting is better than that of the linearone for lower values of χ . However, the linear EoS obtained here are different from thatcorresponds to MIT bag model [20, 21]. The EoS for matter interior to a compact starin modified gravity is predicted here which is non-linear for massive star. In Table- 2, wedisplayed a and b for different anisotropy ( α ) with χ = 1 for a stable stellar configura-tion. The anisotropy lies in the range − . < α < . χ = 1 in a stable stellar model.( ix ) For observed masses of the pulsars namely, Vela X-1, 4U 1820-30, Cen X-3,LMC X-4, SMC X-1, we determine the values of a , b C with χ = 1 stable anisotropicmodels are shown for anisotropy α = 0 .
5. A class of relativistic solutions are obtainedfor different anisotropic pressure inside the star. The predicted radius of the abovepulsars for the parameters are displayed in Table-3. However, varying the values of a , b C , it is possible to obtain stable anisotropic stellar models with different couplingparameter and anisotropy. It is possible to estimate the corresponding radius which liesin the range (10 ∼ km. for a stable neutron star.Thus a class of new relativistic solutions are found in f ( R, T ) gravity with FS ansatzwhich are useful for building stellar models. The precise measurement of radius of aneutron star in future will be useful to accept the modification incorporated in thegravitational action for building stellar models which can dig out information on thematter inside the star at extreme terrestrial condition. It is found that the EoS for acompact object with modified FS ansatz is non-linear. ompact Objects in f ( R, T ) gravity
7. Acknowledgements
SD is thankful to UGC, New Delhi for financial support. AC would like to thankUniversity of North Bengal for awarding Senior Research Fellowship. The authorswould like to thank IUCAA Resource Center, NBU for extending research facilities.BCP would like to thank DST-SERB Govt. of India (File No.: EMR/2016/005734) fora project. [1] M. Sami,
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